{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:VKGGHNQUQMULNPDSAHMELVF32Q","short_pith_number":"pith:VKGGHNQU","canonical_record":{"source":{"id":"1103.2059","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-10T15:33:13Z","cross_cats_sorted":["cs.DM","cs.SI","math.MG"],"title_canon_sha256":"f02c5526833aaa787ff421c765edd3d06ea7fecf77cc143c0474044cf8954f22","abstract_canon_sha256":"17f140710ef388cfe47a5b3e149deafaae0637cab930c0e0b319ee7079daf30d"},"schema_version":"1.0"},"canonical_sha256":"aa8c63b6148328b6bc7201d845d4bbd401520e1cc969b12bcbc9c7bf20f9c12b","source":{"kind":"arxiv","id":"1103.2059","version":8},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2059","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2059v8","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2059","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"VKGGHNQUQMUL","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VKGGHNQUQMULNPDS","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VKGGHNQU","created_at":"2026-05-18T12:26:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:VKGGHNQUQMULNPDSAHMELVF32Q","target":"record","payload":{"canonical_record":{"source":{"id":"1103.2059","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-10T15:33:13Z","cross_cats_sorted":["cs.DM","cs.SI","math.MG"],"title_canon_sha256":"f02c5526833aaa787ff421c765edd3d06ea7fecf77cc143c0474044cf8954f22","abstract_canon_sha256":"17f140710ef388cfe47a5b3e149deafaae0637cab930c0e0b319ee7079daf30d"},"schema_version":"1.0"},"canonical_sha256":"aa8c63b6148328b6bc7201d845d4bbd401520e1cc969b12bcbc9c7bf20f9c12b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:59.819131Z","signature_b64":"D75Hlfgy1QX9Vr8KKvFMopzyAB8Avjjd8h9C0J8tyEiTxGvc4MFYGn689Sd7lLKnaJcU5e0DFuW/WT8cvt1tCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aa8c63b6148328b6bc7201d845d4bbd401520e1cc969b12bcbc9c7bf20f9c12b","last_reissued_at":"2026-05-18T04:00:59.818419Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:59.818419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.2059","source_version":8,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jkeOO9cF5uLMbITI1xsKPkhiAuc0AcPcPzhP1HgBq6IUNCKg18ZePly5jIX7VU0v1QlPkmTdn5Xh+iJVDBDsBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:53:42.035180Z"},"content_sha256":"ebe16d149b6282156d3dcbf9c0065b8ee0ff087ba0fc9939801acbef3a11b410","schema_version":"1.0","event_id":"sha256:ebe16d149b6282156d3dcbf9c0065b8ee0ff087ba0fc9939801acbef3a11b410"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:VKGGHNQUQMULNPDSAHMELVF32Q","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Walk Distances in Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","cs.SI","math.MG"],"primary_cat":"math.CO","authors_text":"Pavel Chebotarev","submitted_at":"2011-03-10T15:33:13Z","abstract_excerpt":"The walk distances in graphs are defined as the result of appropriate transformations of the $\\sum_{k=0}^\\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2059","kind":"arxiv","version":8},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iW5SS8GKXdcgLOkIlmno6HslKkH8gbeCaPR/JB28Zlo/sI7jSV5d2KAcyT5iZlhJm6ndgRGFW7b1Nqt93ux5CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T13:53:42.035556Z"},"content_sha256":"b2e206a5baa42cb813e7b946e9b93a28ffa797622ef91a2d5a44ae10f635bcb9","schema_version":"1.0","event_id":"sha256:b2e206a5baa42cb813e7b946e9b93a28ffa797622ef91a2d5a44ae10f635bcb9"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VKGGHNQUQMULNPDSAHMELVF32Q/bundle.json","state_url":"https://pith.science/pith/VKGGHNQUQMULNPDSAHMELVF32Q/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VKGGHNQUQMULNPDSAHMELVF32Q/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-07T13:53:42Z","links":{"resolver":"https://pith.science/pith/VKGGHNQUQMULNPDSAHMELVF32Q","bundle":"https://pith.science/pith/VKGGHNQUQMULNPDSAHMELVF32Q/bundle.json","state":"https://pith.science/pith/VKGGHNQUQMULNPDSAHMELVF32Q/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VKGGHNQUQMULNPDSAHMELVF32Q/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:VKGGHNQUQMULNPDSAHMELVF32Q","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"17f140710ef388cfe47a5b3e149deafaae0637cab930c0e0b319ee7079daf30d","cross_cats_sorted":["cs.DM","cs.SI","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-10T15:33:13Z","title_canon_sha256":"f02c5526833aaa787ff421c765edd3d06ea7fecf77cc143c0474044cf8954f22"},"schema_version":"1.0","source":{"id":"1103.2059","kind":"arxiv","version":8}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.2059","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"arxiv_version","alias_value":"1103.2059v8","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.2059","created_at":"2026-05-18T04:00:59Z"},{"alias_kind":"pith_short_12","alias_value":"VKGGHNQUQMUL","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_16","alias_value":"VKGGHNQUQMULNPDS","created_at":"2026-05-18T12:26:44Z"},{"alias_kind":"pith_short_8","alias_value":"VKGGHNQU","created_at":"2026-05-18T12:26:44Z"}],"graph_snapshots":[{"event_id":"sha256:b2e206a5baa42cb813e7b946e9b93a28ffa797622ef91a2d5a44ae10f635bcb9","target":"graph","created_at":"2026-05-18T04:00:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The walk distances in graphs are defined as the result of appropriate transformations of the $\\sum_{k=0}^\\infty(tA)^k$ proximity measures, where $A$ is the weighted adjacency matrix of a graph and $t$ is a sufficiently small positive parameter. The walk distances are graph-geodetic; moreover, they converge to the shortest path distance and to the so-called long walk distance as the parameter $t$ approaches its limiting values. We also show that the logarithmic forest distances which are known to generalize the resistance distance and the shortest path distance are a subclass of walk distances.","authors_text":"Pavel Chebotarev","cross_cats":["cs.DM","cs.SI","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-10T15:33:13Z","title":"The Walk Distances in Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.2059","kind":"arxiv","version":8},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ebe16d149b6282156d3dcbf9c0065b8ee0ff087ba0fc9939801acbef3a11b410","target":"record","created_at":"2026-05-18T04:00:59Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"17f140710ef388cfe47a5b3e149deafaae0637cab930c0e0b319ee7079daf30d","cross_cats_sorted":["cs.DM","cs.SI","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-03-10T15:33:13Z","title_canon_sha256":"f02c5526833aaa787ff421c765edd3d06ea7fecf77cc143c0474044cf8954f22"},"schema_version":"1.0","source":{"id":"1103.2059","kind":"arxiv","version":8}},"canonical_sha256":"aa8c63b6148328b6bc7201d845d4bbd401520e1cc969b12bcbc9c7bf20f9c12b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa8c63b6148328b6bc7201d845d4bbd401520e1cc969b12bcbc9c7bf20f9c12b","first_computed_at":"2026-05-18T04:00:59.818419Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:59.818419Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D75Hlfgy1QX9Vr8KKvFMopzyAB8Avjjd8h9C0J8tyEiTxGvc4MFYGn689Sd7lLKnaJcU5e0DFuW/WT8cvt1tCw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:59.819131Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.2059","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ebe16d149b6282156d3dcbf9c0065b8ee0ff087ba0fc9939801acbef3a11b410","sha256:b2e206a5baa42cb813e7b946e9b93a28ffa797622ef91a2d5a44ae10f635bcb9"],"state_sha256":"4689a3649064c39f9732a2f4c0743969ca1ccdbf76affd25ae00bb46da15afa6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BFlVgBsij37sABE9KfzaVDlXbRrH4TU1bCX4eMjjFL9FAFWb3+LqsQHt0th2GkttQOhRjpJlI4V6gabFaXX5BA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T13:53:42.037519Z","bundle_sha256":"cd9af8ec5683edd44b904ad5a700ba00f70cb1e1314739a33309076f57b19555"}}